What does “particle number conservation” mean in condensed matter physics?

Question

What exactly does it imply about a condensed matter system to have particle number conserved or not conserved?

For example, why does the superconducting phase break particle number conservation while the insulating phase does not? What exactly brings about the breaking of this conservation?

Answer 1

Think of writing down the equation for the number operator N in the Heisenberg picture. Now if N commutes with the Hamiltonian H then the time derivative of N is 0, i.e. N is conserved.